A Communication-Efficient Decentralized Newton's Method with Provably Faster Convergence
Abstract
In this paper, we consider a strongly convex finite-sum minimization problem over a decentralized network and propose a communication-efficient decentralized Newton's method for solving it. We first apply dynamic average consensus (DAC) so that each node is able to use a local gradient approximation and a local Hessian approximation to track the global gradient and Hessian, respectively. Second, since exchanging Hessian approximations is far from communication-efficient, we require the nodes to exchange the compressed ones instead and then apply an error compensation mechanism to correct for the compression noise. Third, we introduce multi-step consensus for exchanging local variables and local gradient approximations to balance between computation and communication. To avoid each node transmitting the entire local Hessian approximation, we design a compression procedure with error compensation to estimate the global Hessian in a communication-efficient way. With novel analysis, we establish the globally linear (resp., asymptotically super-linear) convergence rate of the proposed method when m is constant (resp., tends to infinity), where m is the number of consensus inner steps. To the best of our knowledge, this is the first super-linear convergence result for a communication-efficient decentralized Newton's method. Moreover, the rate we establish is provably faster than those of first-order methods. Our numerical results on various applications corroborate the theoretical findings.
Cite
@article{arxiv.2210.00184,
title = {A Communication-Efficient Decentralized Newton's Method with Provably Faster Convergence},
author = {Huikang Liu and Jiaojiao Zhang and Anthony Man-Cho So and Qing Ling},
journal= {arXiv preprint arXiv:2210.00184},
year = {2022}
}